The term “properties of addition” in mathematics refers to a collection of rules that control how two or more integers are added together. These rules apply to algebraic expressions, integers, fractions, and decimals alike. The features of addition allow for the simplification of computations and the efficient solution of challenging issues. The purpose of this essay is to explore more into the addition characteristics and their importance in mathematics.
Mathematical properties of addition refer to the specific rules and conditions that are associated with this arithmetic operation. These properties are also indicative of the closure property of addition. It is worth noting that similar to addition, mathematical properties are defined for subtraction, multiplication, and division, but they can vary depending on the operation. In the case of addition, there are four fundamental mathematical properties that have been defined. These include the commutative property, associative property, distributive property, and additive identity property. In the following sections, we will discuss each of these properties in detail.
The commutative property of addition states that the order of the numbers or integers being added does not affect the sum. This property is also applicable to multiplication. It can be expressed as:
A + B = B + A
Let’s say that A is 8 and B is 2.
8 + 2 = 2 + 8
10 = 10
In this case, we can see that the result, 10, remains the same even after adding the two numbers, 8, and 2, and switching their locations. This illustrates how addition has the commutative property. The word “commute,” which denotes exchanging or shifting locations, is a helpful way to recall this characteristic.
According to the associative property of addition, the addition of three or more numbers together has no effect on the outcome. In other words, no matter how we arrange the addends while adding three or more integers, the total will always be the same. This characteristic can be stated as:
(A + B) = (A + B) + C
Assume that A is 3, B is 6, and C is 9.
LHS equals 3 + (6 + 9) plus A + (B + C)
RHS is equal to (A + B) + C (3 + 6) + 9
18 = 18
The left side is equivalent to the right side, as shown in this example. This supports addition’s associative characteristic. This characteristic applies to multiplication as well. The addends are grouped together and operations using a collection of integers are formed using brackets. This quality can be remembered by the verb “associate,” which denotes the grouping or affiliation with a specific set of people.
Multiplication’s distributive property stands apart from its commutative and associative features. According to this principle, multiplying a number by the sum of two other numbers produces the same results as multiplying the number by each individual number separately and then adding the results. The distributive property of multiplication can be expressed mathematically as:
A = (B + C) = A + (B + C)
A is the monomial factor and (B + C) is the binomial factor in this formula.
Assume that A is 4, B is 7, and C is 9.
LHS is equal to A (B + C) = 4 (7 + 9)
= 4 × 16
RHS is equal to A + B + C = 4 + 7 + 4 + 9.
= 28 + 36
64 = 64
As can be seen in the example above, the outcome is the same on both sides despite allocating A (the monomial factor) to both values of the binomial factor, B and C. Due to the fact that it mixes addition and multiplication operations, the distributive property is essential.
It may be inferred from this feature that every real number has a different real number that, when added to it, produces the same number. The identity element of addition is a special real number that is represented by the zero.
The identity feature of addition is demonstrated by the expressions A + 0 = A or 0 + A = A.
For example, 9 + 0 or 9 + 0 equals 9.
One can ask themselves the question, “What number should be added to the given number to keep the original value unchanged?” to help them remember the identity feature of addition. This question always has a zero response, making it the identity element of addition.
In order to reduce the complexity of 5(2+3) using addition-related principles, we can first apply the distributive property of multiplication over addition, which results in:
5(2+3) = 52 + 53
= 10 + 15
We can see that the number that must be added to 5 in order to get 0 is -5, which is the answer to the missing number in the equation 5 + ____ = 0. Therefore, -5 should be used to fill in the blank:
5 + (-5) = 0
In order to evaluate – (7+2) using properties of addition, we can first simplify the expression enclosed in brackets, which results in:
– (7+2) = – 9
– (7+2) = –7 + (–2) = –9
Consequently, – (7+2) = -9.
Frequently Asked Questions on Properties of Addition
In many mathematical situations, the properties of addition are used to transform the complicated expression into a simple expression.
The commutative property of addition states that even if the order of the addends changes during the addition process, the sum stays the same.
Seven has the additive identity of 0. Since there is only one additive element—zero—the original number’s value is unaffected. Therefore, 7 + 0 = 7.
The distributive property is the one that makes use of both addition and multiplication. In other words, A (B + C) = A B + A C